用稀疏奖励训练老师，密集监督压缩到小模型，提升推理性能

In settings where labeled verifiable training data is the binding constraint, each checked example should be allocated to the model and reward density where it is most informative. We identify a reward-density principle that governs this allocation: sparse sequence-level reward is most useful on models that can explore and discover better behavior, while dense token-level teacher supervision is better suited for compressing that behavior into a smaller deployment model. The principle yields a simple allocation rule: use scarce labeled data upstream on the strongest available teacher, then transfer the reward-shaped behavior downstream as dense supervision. We evaluate this rule through a four-stage workflow -- teacher RL, forward-KL warmup, on-policy distillation, optional post-bridge student RL -- on verifiable math with Qwen3 and Llama models. At fixed Qwen3-1.7B deployment-student size, an RL-improved 8B teacher distilled through the dense bridge outperforms direct GRPO on the same student ($79.3\%$ vs.\ $75.9\%$ on MATH; $25.2\%$ vs.\ $19.8\%$ on AIME~2024, avg@16), while transfer from the same teacher \emph{before} RL underperforms. A component ablation confirms that each stage is load-bearing: replacing the RL-improved teacher with a raw teacher costs $7.8$ MATH points, removing the forward-KL warmup costs $1.7$, and removing on-policy distillation costs $3.3$. The teacher-quality ordering -- raw-teacher transfer $<$ direct GRPO $<$ RL-teacher transfer -- replicates on Llama-3.1-8B-Instruct with a Llama-3.3-70B-Instruct teacher. The operational lesson is to avoid spending scarce labeled data on the least prepared policy: use sparse reward for teacher-side discovery, dense transfer for student compression, and student-side sparse reward only after the bridge.